As with any other sport, it is interesting to study and apply simple and complex notions of physics in tennis. Concepts such as moments of inertia, elastic collisions and momentum are a constant part of the match. The technical preparation of tennis players also takes physics into account and most of the training techniques are based on it thanks to concepts such as point of impact, balance, inertia and masses.
The tennis racket
In sports where a tool is used to hit a ball, the racket (see figure 1) is usually six times heavier than the ball and about one sixth the weight of the player’s arm. The tennis ball weighs 57 grams. One arm weighs approximately 2 kg. The ideal weight of the racket is around 340 grams. What is the reason for this numerical relationship? The weight of the racket tends to slow down the speed of the arm especially when the weight of the tool exceeds this ratio. When the racket hits the ball, its speed also depends on its weight. At the same speed, a ball hit by a heavier tool will be faster. But if the weight increases too much, the speed of the racket decreases and the hit is less effective. It gets to the point that a 500 gram racket produces the same result as a 600 gram racket. Also, a 200 gram racket produces twice as efficient results as a 100 gram racquet, but it is not possible to move the 100 gram racket at twice the speed of the 200 gram one. Therefore, the ideal weight of the racket is around 340 grams. The ratio of 1/6 between ball and racket and between racket and arm is the best.
Figure 1: The racket is considered to be a natural extension of the arm
With regard to the impact between the racket and the ball, when two objects collide, they generate a force that changes their state of motion. The force necessary to modify the state of motion of a body, that is to accelerate or decelerate it, depends on its mass, according to the following formula:
“F” is the force that modifies the state of motion;
“m” is the mass;
“a” is the acceleration.
You’d need twice the force to accelerate or decelerate a body that has twice the mass of the other. The collision between two moving bodies determines the release of a certain amount of kinetic energy, which depends on their mass and their speed (kinetic energy is equal to half the mass times the speed squared). Since the ball and the strings are elastic materials, upon impact they are able to conserve a certain amount of kinetic energy, transforming it into elastic energy which can then be returned to the bodies upon impact. A part of the accumulated elastic potential energy is inevitably lost and it is transformed into vibrations and heat due to friction; in the case of the strings-ball impact, the ball loses about 45% of its elastic energy. This loss is required by the rules of tennis and prevents the ball from traveling too fast and be too dangerous. Generally speaking, there is no perfect racket, but a player must always try out several rackets before finally choosing the model that he will use during tournaments.
The movements of the tennis player
Nowadays, to improve performance, tennis players study the different behaviors of any action. With the help of their trainers, players perform kinematic and dynamic analysis to study the relationships between the forces acting on a body and its motion. In tennis, the most studied gesture is the serve, because it is a shot performed with a series of precise movements and does not depend on the opponent or other agents of the game in progress (see figure 2). The serve is a very important shot because it starts the game but also offers the possibility of winning the point immediately. It is also a double-edged sword because, with two wrong serves, the player loses the point due to a double fault. The power of the shot is linked to the thrust of the legs, the rotation of the trunk, the action of the shoulders and the rotation of the arm around it, together with the position of the racket with a variable distance from the rotation axis, according to the formula:
With this formula it is stated that the inertia of the body is greater the greater the distance from the axis of rotation.
Figure 2: The tennis serve can be an immediate winning move
Newton’s three laws govern a tennis match
Physics, together with Mathematics, are the backbones of any sporting event. In a game of tennis, Newton’s three laws can be constantly observed:
the principle of inertia states that a body remains in its state of rest or uniform rectilinear motion until an external force intervenes and modifies this state;
the second principle of dynamics states that a force acting on a body determines an acceleration which is directly proportional to the intensity of the force and inversely proportional to the mass of the body;
the principle of action and reaction states that, if a force acts on a body, there is another body on which an equal and opposite force acts.
The Speed of the ball after the impact
The speed of the ball after impact with the racket strings is given by:
v (inp) is the speed of the incoming ball;
V (r) is the speed of the racket head at the point of impact;
and (A) is the apparent refund coefficient.
With a string tension of 280 N, the apparent restitution coefficient is about 0.4. If the string tension is reduced to 224 N, the apparent return coefficient increases by 7%, becoming 0.433. The result is a 3% increase in ball speed. A racket with a tension of 230 N produces, in practice, fewer errors, while a racket of 180 N produces more errors.
The “sweet spot”
The sweet spot is a point on the racket strings that is stressed by the ball upon impact. This is where you have the best rebound on the stringbed. A tennis racket has two sweet spots. If the ball is hit in one of these two areas, the forces transmitted from the racket frame to the arm are extremely small and the vibrations are not detected by the player. These points cause a certain vibration of the strings around 100 Hz (for a flexible frame) and around 185 Hz (for a rigid frame). The impact of the ball on the racket is extremely short, only 5 milliseconds. The sweet spot does not coincide with the point where the ball receives the maximum thrust. A less rigid and more flexible racket produces less vibrations and is softer and less damaging to the joints. Furthermore, the vibration frequency of a racket depends on the stiffness of the frame (see figure 3). A rigid racket vibrates at 180Hz or more, while a flexible one at 140Hz or less. There is also another critical point, the “dead spot”, in which the energy is not returned entirely from the racket to the ball itself because the actual mass of the racket at that point is equal to the mass of the ball. The various points are placed in different places and are not fixed for every type of racquet. Raising the weight of the racquet will lower or raise the positions of those points. One might think that the best spot on the racket to hit the ball with is the one at the center, but that’s not the case. Obviously, to prevent the handle from rotating in the hand, the impact must occur along the vertical axis as an extension of the handle. In the center of the strings the kickback on the hand is minimal but a lot of power is lost due to the production of vibrations on the arm. The slightest vibration occurs when the blow occurs just above the center of the strings. The maximum power, on the other hand, is just below.
Figure 3: The various points of impact produce different vibrations
A single biological tool
Every movement in tennis involves a close connection between the arm, the hand and the racket. The latter represents the natural extension of our arm and is a tool that must be managed with awareness and precision as if it were part of the human body. To perform a correct movement, time, space and acceleration must be perfectly correlated. In other words, the correct distance to the optimal impact point must be calculated. In doing so, the necessary space must be created for the action of the arm and racket for optimal impact and perfect timing. In this space-time context, we try to make the racket move in the space created, thus providing optimal acceleration for maximum energy transfer to the ball. Timing summarizes all the action and is a correct combination of time, space, and acceleration. The arm-racket movement allows fast movement in a given space and follows a trajectory that goes from top to bottom with an oval shape. This action is also influenced by the force of gravity and the tennis player must be able to combine this force with that produced by his limb, to maximize the energy produced. The racket moves following an acceleration and, at the moment of maximum loading, it is ready to hit the ball. The racket moves from top to bottom thanks to the force of gravity and the tennis player must not abruptly block this action, which would cause a zeroing of all the energy created.
Swingweight is the moment of inertia of the racket, and can be defined as “the resistance to rotation with respect to an axis”. The moment of inertia of a rotating body is the result of the sum of all the mass points for their distance from the rotation axis to the square (see Figure 4). With an example we explain the concept. We have the following two rackets:
first racket weighing 340 grams with a weight of 10 grams in the center;
second racket weighing 340 grams with two 5 gram weights at the ends.
The two rackets, therefore, both weigh 350 grams and have the same balance point, but the second racket feels heavier. It has a harder time starting its rotation, but once it starts its speed will be faster, even if it seems heavier. This principle is known as “swingweight” or “moment of inertia” and greatly affects the handling of a racquet. The higher this value is, the heavier and less manageable the racket seems, but it transfers more power and speed to the ball. For this reason, the limbs must have a more robust musculature. The formula for its calculation is as follows:
m: mass added to the racket;
d: distance of the added weight from the bottom of the frame.
The result must be added to the initial SW. If you add weight to the handle, the SW does not increase but only varies the overall weight and handling of the racquet.
Figure 4: the concept of swingweight
The topspin is one of the most important strokes in the game of tennis. The action consists in hitting the ball from the bottom up with a quick flick of the wrist. You need to give the ball a rotational movement with the racket to activate the Magnus effect (just like in football), in order to make the ball rotate quickly in the fluid (air). A rotating body in a fluid drags the layer of fluid immediately in contact with it and the latter, in turn, drags another layer of fluid forming many layers of fluid rotating on different concentric circles. Depending on the rotation speed, the translation effects of the ball can be different.
The type of court
The places where matches are held can have different surfaces. Grass, dirt and concrete produce very different game results. On a physical level, the behavior of the ball is different. A first feature is provided by the rebound theory which provides a physical system in which a spherical body is equipped with a horizontal speed produced by the hit of the racket (vx1) and a vertical speed caused by the force of gravity (vy1), as shown in figure 5. After the impact with the surface, for a determined angle of incidence, the two speeds are reduced (vx2 and vy2) and the ball (deforming temporarily and heating up) loses part of its energy also based on the surface on which it bounces.
Figure 5: the bounce of the ball on the ground
For a few fractions of a second the ball also begins to slide on the court, covering a very small distance (D) whose duration depends on the friction of the surface itself. After this transition phase, the ball resumes the rotation motion and rises for the next bounce. The rebound, of course, is different whether the shot has topspin or not. Figure 6 shows a simulation of striking on clay ground (red graph) and on grass (green graph). The first chart refers to a hit without topspin, the second to a hit with topspin. Without topspin the ball, thrown at 130 km / h, would reach the baseline at a speed of 14% less. With a topspin of around 4000 RPM, the ball reaches the baseline (and after the bounce) at the same speed and after the same time. There is only a difference of about 50 centimeters in height in favor of clay.
Figure 6: the different types of bounce on clay and grass
In the case of a strongly rotating ball, the tangential speed of the top of the ball is greater than the overall horizontal speed at the moment of impact with the surface. During the impact, the upper tangential speed is reduced to equal the horizontal one, then the ball resumes rolling and starts up again after suffering the effect of the friction of the surface. Terrain slows flat shots much more than spin shots.
Tennis is a sport in which the concepts of physics and mathematics can be applied at any time. The differences in terrain and strokes make games unpredictable and tennis players study these concepts thoroughly to win their games. There are hard courts that slow flat shots but speed up spinning shots. The game becomes almost a personal matter among the athletes. Atmospheric conditions can also affect shots: humidity, temperature and pressure are parameters to be taken into consideration. One thing is certain: a tennis ball thrown at maximum speed has a very large kinetic energy that could even be lethal to humans.